Finding the Mass of Jupiter Using its moons.
By: Kiran Chand, Ng Keen Yung, Lim Zhong Zhi, Anbarasan Subramaniyan
Mentor: Mr Tan Hoe Teck
School: School of Science and Technology
Programme: Interdisciplinary Research Studies Proposal
Category: Space Science
The mass of Jupiter using one of its moons’ orbit can be found using Kepler's Law and Newton's Second Law.
-Needs Kepler's 3rd law and Newton’s 2nd law as there is no mass in Kepler's 3rd law.
----Kepler's third law:
- P^2 = a^3
- a = the semi-major axis of an elliptical or circular orbit ( a circle is just an ellipse with 0.00000..... eccentricity)
- P = time for 1 complete orbit around the sun
Units need to be consistent. (e.g: Units used for the semi-major axis are astronomical units, then answer must be in years. )
Need a scientific calculator
---Sir Isaac Newton's Law of Universal Gravitation:
- F= GMm//r^2 where
- G = the universal gravitational constant = 6.67 x 10^-11 meters^2/kg^2
- M = the mass of Jupiter
- m = the mass of the moon you choose
- r = a = the distance of the center of the moon you choose from the distance of Jupiter’s center
Ganymede is one of the moons that we can use for our calculations. 2 more of Jupiter's moons can be chosen to confirm our calculations. Even if you get slightly different answers for Jupiter's mass as long as the answers have the same order of magnitude power of 10" your answers are probably correct.
1)We start by determining the mass of the Jupiter’s moon
2)Issac Newton's Law of Universal Gravitation tells us that the force of attraction between two objects is proportional the product of their masses divided by the square of the distance between their centers of mass. To obtain a reasonable approximation, we assume their geographical centers are their centers of mass.
3)Because we know the radius of the Jupiter’s moon(Ganymede)we can use the Law of Universal Gravitation to calculate the mass of Ganymede in terms of the gravitational force on an object (its weight) at Ganymede’s surface using the radius of Ganymede as the distance.
4)We also need the Constant of Proportionality in the Law of Universal Gravitation, G.
5)Knowing the mass and radius of the Earth and the distance of the Earth from the sun, we can calculate the mass of the Jupiter again by using the law of universal gravitation.
6)The gravitational attraction between the Ganymede and the Jupiter is G times the Jupiter’s mass times the Ganymede’s mass, divided by the distance between the Ganymede and the Jupiter squared.
7)This attraction must be equal to the centripetal force needed to keep the Ganymede in its (almost circular) orbit around the Jupiter
8)The centripetal force is the Ganymede’s mass times the square of its speed divided by its distance from the Jupiter
9)By astronomically determining the distance to the Jupiter, we can calculate the Ganymede’s speed around the sun and hence the Jupiter’s mass.
Risk assessment skill:
List/identify the hazardous chemicals, activities, or devices that will be used
1.Identify and assess the risks involved?
Ans:Using the telescope (without adult supervision and using it carelessly).
2.Identify and assess the safety risks involved?
Ans:Seeing the moon and it is too bright and it will cause damage to the eyes
3.Describe the safety procedures?
Ans:We will ask an adult to set the magnification and the angle properly so we will not see anything too bright or harmful to us.We might also consider handling the telescopes with care as some of them are fragile.
Some measures We may consider
As science centre observatory is a free to the public place, there will be visitors who may be there to observe the stars and they will use the telescopes that we might be using. Therefore, we might get deprived of taking pictures using both the telescopes at the same time. We might want to wait till the all the people have seen through the telescope and then we will use the telescope and take pictures. Then we might also take note of not using the telescope ourselves as we are depriving other visitors from using the telescope.
Some problems we might encounter
When we are taking pictures through the telescope the clouds may block the view of the telescope. Therefore, we might take pictures whenever available/best. Our eyes might be damaged if we accidentally point the telescope in the wrong direction. We might also handle the telescope wrongly.Photo importing problems may occur.